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The modified Kawahara equation also called Korteweg-de Vries (KdV) equation of fifth-order arises in shallow water wave and capillary gravity water waves. This study is based on the generalized Riccati equation mapping and modified the F-expansion methods. Several types of solitons such as Bright soliton, Dark-lump soliton, combined bright dark solitary waves, have been derived for the modified Kawahara equation. The obtained solutions have significant applications in applied physics and engineering. Moreover, stability of the problem is presented after being examined through linear stability analysis that justify that all solutions are stable. We also present some solution graphically in 3D and 2D that gives easy understanding about physical explanation of the modified Kawahara equation. The calculated work and achieved outcomes depict the power of the present methods. Furthermore, we can solve various other nonlinear problems with the help of simple and effective techniques.

This study introduces innovative analytical solutions for the $(2+1)$-dimensional nonlinear Jaulent–Miodek ($𝕁𝕄$) equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of $𝕁𝕄$ waves. Within the realm of nonlinear fluid dynamics, the $𝕁𝕄$ equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the $𝕁𝕄$ equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the $𝕁𝕄$ equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the $(2+1)$-dimensional $𝕁𝕄$ equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field.

The unsmooth boundary has a great influence on the solitary wave form of a nonlinear wave equation. It this work, we for the first time ever propose the fractal regularized long-wave equation which can describe the shallow water waves under the unsmooth boundary (such as the fractal seabed). The fractal variational principle is established and is proved to have a strong minimum condition by the He–Weierstrass theorem. Then, the solitary wave solution is obtained by the fractal variational method which can reduce the order of differential equation and obtain the optimal solution by the stationary condition. Finally, the impact of the unsmooth boundary on the solitary wave is presented. It shows that the fractal order can affect the wave morphology, but cannot affect its peak value. The finding in this paper is important for the coast protection and expected to bring a light to the study of the fractal theoretical basis in the geosciences.

Solitary wave motion, which arises in various fields of science and engineering such as hydrodynamics, quantum mechanics, plasma physics, and particle physics, is a nonlinear and weakly dispersive phenomenon generally modeled by a third-order nonlinear partial differential equation called the Korteweg–de Vries (KdV) equation. Our aim is to obtain a numerical solution of the governing KdV equation for shallow water waves using a robust and efficient numerical scheme called differential quadrature method (DQM), which has potential applications in solving higher-order differential equations, generally occurring in solid mechanics. One of the most promising advantages of DQM over standard numerical schemes, such as finite difference, finite element, and finite volume methods, is that it provides a very accurate and stable solution for a partial differential equation (PDE) using very few grid points. In fact, the solution becomes unstable if a large number of grid points are used in contrast to that of the standard numerical schemes, where solution becomes more accurate when spacing between grid points is reduced. This unique property of DQM makes it a fast algorithm for solving PDEs. Standard polynomial-based DQM with unequally spaced Gauss– Lobatto–Chebyshev points are used here for the simulation studies. The suggested numerical scheme is used for solving test problems involving a single solitary wave as well as interactions of two and three solitary waves. To check the accuracy of the newly applied method, the error norms, L₂ and L_∞, have been calculated. A fuzzy differential quadrature has been developed to solve the fuzzy KdV equation, which is formulated by considering the parameters associated with the nonlinear and dispersive terms of the KdV equation as fuzzy variables. The advantage of fuzzy differential quadrature is that we can reach the classical DQM solution from fuzzy DQM.